Our coalgebraic characterization of linear bisimulation proceeds as follows. We
define a \(K\) -linear relation on a semimodule \(V\)  over the semiring \(K\)  as the
kernel of a \(K\) -linear map \(f:V \rightarrow W\)  with domain \(V\) . Clearly, this is an
equivalence relation which is a congruence. In fact any relation \(R\)  on \(V\)  can
be turned into a \(K\) -linear relation by considering the smallest congruence
\(R^\ell \)  containing \(R\)  and taking \(f\)  to the be canonical map \(V \rightarrow V/R^\ell \) 
that sends each element to its congruence class. A \(K\) -linear bisimulation on a
weighted automaton is then defined as a \(K\) -linear relation which preserves the
output map of the weighted automaton and is invariant under the transition map
as in [1]}. We show that this approach leads to all the
results of [1]} regarding the functor \(\mathcal {L}\)  and weighted
language equivalence although the proofs are new as we cannot assume the field
properties of \(K\) , in particular the existence of the additive inverse.
Table REF  is a summary of the correspondence between the key concepts
in [3]}, [1]} and the present work.
<TABLE>